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We study the superconductor-insulator transition of Bose-Hubbard models with finite-range interactions. Commensurability of the charge distribution with the underlying lattice leads to a richly structured phase diagram. In addition to the lobes of insulating phase characterized by integer fillings, we find--for finite-range interactions--lobes with rational filling factors. At low temperatures we can investigate the phase transition by mapping the model onto a XXZ spin-1/2 Heisenberg model.
Bruder et al. (Fri,) studied this question.